A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density

Abstract

Abstract. Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole n -dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space ofḂ

A Lagrangian approach for the compressible Navier-Stokes equations

“…Following our recent paper [14] dedicated to the incompressible densitydependent Navier-Stokes equation, and older works concerning the compressible Navier-Stokes equations (see [21,22,23]), we here aim at solving System (0.1) in the Lagrangian coordinates. The main motivation is that the mass is constant along the flow hence, to some extent, only the (parabolic type) equation for the velocity has to be considered.…”

“…• The regularity condition over the density is stronger than that for density-dependent incompressible fluids (see [14]). In particular, in contrast with incompressible fluids, it is not clear that combining Lagrangian coordinates and critical regularity approach allows to consider discontinuous densities.…”

Fourier Analysis Methods for the Compressible Navier-Stokes Equations

“…The term with u L goes to 0 for T tending to 0 with a speed of convergence that may be described according to (19). To handle the second term, we observe that u n+1 satisfies…”

New Maximal Regularity Results for the Heat Equation in Exterior Domains, and Applications

“…The model may thus be reformulated equivalently in Lagrangian variables (see e.g. our recent work [4] in the slightly different context of incompressible flows). This is obviously of interest to investigate free boundary problems.…”

“…≤ C( f L 1 (0,T ;Ḃ s p,1 (R n )) + u 0 Ḃ s p,1 (R n ) ), (4) where the homogeneous Besov semi-norm that is used in the above inequality is defined by…”